scholarly journals Some Formulas and Examples for Two-dimensional Stress Problems Solved only by Harmonic Function

1956 ◽  
Vol 22 (119) ◽  
pp. 462-468
Author(s):  
Yoitiro TAKEUTI
Keyword(s):  
Harmonic Function ◽  
Two Dimensional

The Exact Deformation Theory of Two-Dimensional Dodecagonal Quasicrystal Shaft

2011 ◽  
Vol 213 ◽  
pp. 276-280
Author(s):  
Bao Sheng Zhao ◽  
Ying Tao Zhao ◽  
Yang Gao
Keyword(s):  
Harmonic Function ◽  
Deformation Theory ◽  
Ad Hoc ◽  
Reverse Direction ◽  
Refined Theory ◽  
Two Dimensional ◽  
Surface Loading ◽  
Circular Shaft ◽  
Cylindrical Coordinate ◽  
Classical Elasticity Theory

Cheng’s refined theory is extended to investigate torsional circular shaft of two-dimensional dodecagonal quasicrystal (2D dodecagonal QCs), and Lur’e method about harmonic function is extended to harmonic function in the respective cylindrical coordinate. The exact deformation of torsional circular shaft of 2D dodecagonal QCs under reverse direction surface loading is proposed on the basis of the classical elasticity theory and stress-displacement relations of 2D dodecagonal QCs, and the exact deformation theory provides the solutions about torsional deformation of a circular shaft without ad hoc assumptions. Exact solutions are obtained for circular shaft from boundary conditions. Using Taylor series of the Bessel functions and then dropping all the terms associated with the higher-order terms, we obtain the approximate expressions for circular shaft of 2D dodecagonal QCs under reverse direction surface. To illustrate the application of the theory developed, one example is examined.

scholarly journals ON BOUNDED-TYPE THIN LOCAL SETS OF THE TWO-DIMENSIONAL GAUSSIAN FREE FIELD

2017 ◽  
Vol 18 (3) ◽  
pp. 591-618 ◽  
Author(s):  
Juhan Aru ◽  
Avelio Sepúlveda ◽  
Wendelin Werner
Keyword(s):  
Harmonic Function ◽  
Connected Domain ◽  
Free Field ◽  
Mean Value ◽  
Two Dimensional ◽  
Gaussian Free Field ◽  
Simply Connected Domain ◽  
The Mean ◽  
Conformal Loop Ensemble ◽  
Simply Connected

We study certain classes of local sets of the two-dimensional Gaussian free field (GFF) in a simply connected domain, and their relation to the conformal loop ensemble$\text{CLE}_{4}$and its variants. More specifically, we consider bounded-type thin local sets (BTLS), where thin means that the local set is small in size, and bounded type means that the harmonic function describing the mean value of the field away from the local set is bounded by some deterministic constant. We show that a local set is a BTLS if and only if it is contained in some nested version of the$\text{CLE}_{4}$carpet, and prove that all BTLS are necessarily connected to the boundary of the domain. We also construct all possible BTLS for which the corresponding harmonic function takes only two prescribed values and show that all these sets (and this includes the case of$\text{CLE}_{4}$) are in fact measurable functions of the GFF.

Note on the slow motion of fluid

1936 ◽  
Vol 32 (4) ◽  
pp. 598-613 ◽  
Author(s):  
W. R. Dean
Keyword(s):  
Harmonic Function ◽  
Viscous Liquid ◽  
Stream Function ◽  
Slow Motion ◽  
Normal Derivative ◽  
Great Distance ◽  
Sharp Edge ◽  
Two Dimensional ◽  
Shearing Motion

In this paper we consider the slow two-dimensional motion of viscous liquid past a sharp edge projecting into and normal to the undisturbed direction of the stream. The liquid is supposed bounded by rigid planes represented by ABCDE in Fig. 1, and, apart from the disturbance caused by the projection, is assumed to be in uniform shearing motion. The stream function is then a bi-harmonic function that must vanish together with its normal derivative at all points of the boundary, and must be proportional to y2 at a great distance from the projection.

Spectrophotometric measurements of objective-prism spectra

1966 ◽  
Vol 24 ◽  
pp. 118-119
Author(s):  
Th. Schmidt-Kaler
Keyword(s):  
Progress Report ◽  
Two Dimensional ◽  
Objective Prism ◽  
Made In

I should like to give you a very condensed progress report on some spectrophotometric measurements of objective-prism spectra made in collaboration with H. Leicher at Bonn. The procedure used is almost completely automatic. The measurements are made with the help of a semi-automatic fully digitized registering microphotometer constructed by Hög-Hamburg. The reductions are carried out with the aid of a number of interconnected programmes written for the computer IBM 7090, beginning with the output of the photometer in the form of punched cards and ending with the printing-out of the final two-dimensional classifications.

Introductory paper

1966 ◽  
Vol 24 ◽  
pp. 3-5
Author(s):  
W. W. Morgan
Keyword(s):  
Line Intensity ◽  
Classification Scheme ◽  
Two Dimensional ◽  
Spectral Classification ◽  
Dimensional Array ◽  
Rr Lyrae ◽  
Metallic Line ◽  
Introductory Paper ◽  
Parameter Varying ◽  
Definition Of

1. The definition of “normal” stars in spectral classification changes with time; at the time of the publication of theYerkes Spectral Atlasthe term “normal” was applied to stars whose spectra could be fitted smoothly into a two-dimensional array. Thus, at that time, weak-lined spectra (RR Lyrae and HD 140283) would have been considered peculiar. At the present time we would tend to classify such spectra as “normal”—in a more complicated classification scheme which would have a parameter varying with metallic-line intensity within a specific spectral subdivision.

A one-dimensional self-gravitating stellar gas

1966 ◽  
Vol 25 ◽  
pp. 46-48 ◽  
Author(s):  
M. Lecar
Keyword(s):  
Gravitational Field ◽  
Phase Space ◽  
Motion Picture ◽  
Space Distribution ◽  
Two Dimensional ◽  
One Dimensional ◽  
Phase Space Distribution ◽  
Dimensional Phase Space ◽  
The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

Sign in / Sign up

Export Citation Format

Share Document